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Consider the Dirichlet problem of the following form: Let $D$ be a bounded, connected open set in $R^d$ and $\partial D$ its boundary. Given any continuous function $f$ defined on the boundary $\partial D$, one needs to find a function $u$ which is continuous on $\bar{D }$, equal to $f$ on $\partial D$, and harmonic in $D$. I was reading some lecture notes on Brownian motion and there was a proof that if $D$ satisfies the cone condition, then the Dirichlet problem has a solution of the form $$u(x)=E_x[f(B_{\tau_D})],$$ where $$\tau_D=\inf\{t>0: Β_t \not \in D\}.$$ I think that in the case $d=2$, if $D$ is a Jordan Domain, then the Dirichlet problem always has a solution, even if $D$ does not satisfy the cone condition.

To see this, consider a conformal mapping $\phi$ from the unit disk $\Delta$ to $D$. From Caratheodory's theorem it extends continuously and injectively on $\bar{\Delta}$. Now $f\circ \phi$ is continuous on $\partial\Delta$ and thus has a harmonic extension $F$ on $D$. But then $F\circ \phi^{-1}$ solves the Dirichlet problem.

My question is the following: is $E_x[f(B_{\tau_D})]$ a solution to the Dirichlet problem in this case?

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    The solution of the Dirichlet problem via Brownian motion is a classical result due to Kakutani from 1944. It is covered in many textbooks and you will find lots of write-ups using a simple Google search. And yes, it works for Jordan domains, no matter their geometry. – 2017-06-04
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    @LukasGeyer Can you give me a reference? – 2017-06-05

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The book Karatzas and Shreve, Brownian Motion and Stochastic Calculus, has an extensive treatment of the solution of the Dirichlet problem by Brownian motion in chapter 4. The cone condition is only a sufficient, not a necessary condition for this to apply. In fact, Proposition 2.7 in that chapter shows that whenever $f$ is bounded and Brownian motion exits the domain almost surely in finite time (both automatically satisfied for bounded domains), then any bounded solution to the Dirichlet problem (again, for bounded domains all solutions are automatically bounded) has the representation you give as $u(x) = E_x[f(B_{\tau_D})]$. It also gives a bunch of sufficient conditions on regularity of the boundary. In particular, problem 2.16 shows that a boundary point of a planar domain is regular if it can be accessed by an arc from the complement of $D$, which is always true for Jordan domains.